IODS course project
Santeri Rouhinen
- Chapter 4. Clustering and classification
- Analysis part
- 2. Load Boston data from MASS package
- 3. Graphical overview of the data.
- 4. Standardize the dataset. Create crime rate categorical variable. Split data into training and data parts.
- 5. Fit linear discriminant analysis on the training set.
- 6. Predict the classes with the LDA model.
- 7. Start again with the Boston dataset. Test what number of clusters to run K-means clustering with would make sense.
- Super bonus, 3D plots.
Chapter 4. Clustering and classification
Analysis part
Boston data has 506 observations and 14 variables. It has information about housing in suburbs of Boston.
Description of variables in the Boston dataset:
crim - per capita crime rate by town.
zn - proportion of residential land zoned for lots over 25,000 sq.ft.
indus - proportion of non-retail business acres per town.
chas - Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox - nitrogen oxides concentration (parts per 10 million).
rm - average number of rooms per dwelling.
age - proportion of owner-occupied units built prior to 1940.
dis - weighted mean of distances to five Boston employment centres.
rad - index of accessibility to radial highways.
tax - full-value property-tax rate per $10,000.
ptratio - pupil-teacher ratio by town.
black - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.
lstat - lower status of the population (percent).
medv - median value of owner-occupied homes in $1000s.
2. Load Boston data from MASS package
# Load required libraries
library(ggplot2)
library(dplyr)
library(corrplot)
library(GGally)
library(tidyr)library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectdata('Boston')
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.003. Graphical overview of the data.
pairs(Boston)
# print the correlation matrix
cor_matrix<-cor(Boston)
cor_matrix %>% round(digits = 2)## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00# Histograms of the variables
Boston %>%
gather(key=var_name, value = value) %>%
ggplot(aes(x=value)) +
geom_histogram() +
facet_wrap(~var_name, scales = "free_x")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Create color ramp from dark blue to white to red.
colorVector <- c("blue", "white", "red")
# visualize the correlation matrix
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6, col = colorRampPalette(colorVector)(200))
The histogram plot shows that much of the data does not look like gaussian normally distributed data. Most of the data variables has a high kurtosis, or has two peaks. The correlation plot shows that there are many high positive correlations between different variables: like industy and NO2 gas level and tax revenue; property taxes and access to radial highways. There are some strong negative correlations ones also like median value of owner-occupied homes and lower status of the population; and between distances to five Boston employment centres and proportion of owner-occupied units built prior to 1940.
4. Standardize the dataset. Create crime rate categorical variable. Split data into training and data parts.
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865# change the object to data frame from matrix type.
boston_scaled <- as.data.frame(boston_scaled)
sd(boston_scaled$crim)## [1] 1# Now all the variables have zero mean and SD of 1. This is recommended for clustering.
# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c("low", "med_low", "med_high", "high"))
# look at the table of the new factor crime. The bins have either 127 or 126 elements.
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# Select 80 % of the data and 20 % of the data and split them into separate datasets.
# number of rows in the Boston dataset
n <- nrow(Boston)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create training set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)5. Fit linear discriminant analysis on the training set.
# linear discriminant analysis. The . means all of the variables.
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object. Note that the first linear discriminant explains 95 % of the model, and the third one only 1.2 %.
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2301980 0.2623762 0.2450495 0.2623762
##
## Group means:
## zn indus chas nox rm
## low 0.91525712 -0.8908605 -0.10299142 -0.8483863 0.4295002
## med_low -0.08334565 -0.2951311 -0.08661679 -0.5872960 -0.1323387
## med_high -0.39109146 0.1689534 0.28443258 0.3830397 0.1095356
## high -0.48724019 1.0170298 -0.12375925 1.0380435 -0.4033470
## age dis rad tax ptratio
## low -0.8271561 0.8408270 -0.7015464 -0.7233055 -0.43441270
## med_low -0.3418616 0.4098612 -0.5495709 -0.4993659 -0.06704845
## med_high 0.4349117 -0.3920332 -0.4470800 -0.3546549 -0.34478590
## high 0.8146248 -0.8527739 1.6390172 1.5146914 0.78181164
## black lstat medv
## low 0.3819121 -0.757191246 0.497525719
## med_low 0.3125279 -0.151112125 -0.002233548
## med_high 0.0562525 -0.006157674 0.189950189
## high -0.7998711 0.928770666 -0.702925805
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.091597782 0.72582560 -1.08552595
## indus 0.075032287 -0.20080401 0.50705166
## chas -0.136066170 -0.11949253 -0.05054510
## nox 0.373836580 -0.84617639 -1.24604452
## rm -0.153705564 -0.11732365 -0.17308915
## age 0.062845066 -0.30311814 -0.04593389
## dis -0.134670630 -0.27855764 0.44857222
## rad 3.728089299 0.81545803 0.06238177
## tax -0.007431938 0.08332912 0.32549723
## ptratio 0.108544816 0.01511329 -0.33385861
## black -0.091260534 0.06675804 0.11447525
## lstat 0.243331995 -0.22221458 0.32180038
## medv 0.202650958 -0.44362332 -0.10349837
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9606 0.0291 0.0104# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results. Both lines need to be run at the same time. High class of crime rate looks to cluster pretty well with only a few medium high class elements in it, and only one high class element being at LD1 value of ~0.
plot(lda.fit, col = classes, pch = classes, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
6. Predict the classes with the LDA model.
# Saving of the crime variable in a vector was done in step 5. Remove the crime variable from test data not necessary either.
# test <- dplyr::select(test, -crime)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 22 12 0 0
## med_low 4 13 3 0
## med_high 0 8 16 3
## high 0 0 0 21The categorization of the low and especially high classes of properties was quite successful. The medium low and medium high classes were only about 60 % correct. There was more confusion about the low crime rate properties compared to the high crime rating properties. Having shifting values depending on different runs of the script is annoing.
7. Start again with the Boston dataset. Test what number of clusters to run K-means clustering with would make sense.
# Reload Boston dataset.
data("Boston")
# center and standardize variables
boston_scaled <- scale(Boston)
# change the object to data frame from matrix type.
boston_scaled <- as.data.frame(boston_scaled)
# Calculate the Euclidean distances between observations.
dist_eu <- dist(boston_scaled)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# K-means clustering
km <-kmeans(boston_scaled, centers = 3)
# plot the Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)
# Determening how many clusters to have using within cluster sum of squares (WCSS). Set a seed to make iterations give the same result.
set.seed(123)
# determine the number of clusters using the total within sum of squares (twcss)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
# visualize the results. Two or 3 seems reasonable. For visualization purposes 2 is handier so lets go with that.
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <-kmeans(boston_scaled, centers = 2)
# plot the normalized Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)
# Tried to use ggpairs but that didn't manage to color with the clusters. One does get the correlation values and distributions with ggpairs though.
ggpairs(boston_scaled)
Per capita crime rate by town (crime) is correlated most strongly with accessibility to radial highways (rad, 0.63), full-value property-tax rate (tax, 0.58), and lower status of the population (lstat, 0.46). Rad and tax are very highly correlated (0.91), and rad and tax are both correlated somewhat strongly with lstat (~.45). Since the rad and tax are so highly correlated it is hard to disentangle what encourages crime in the area, good connections or wealth as measured by taxes. I would have imagined that lower status would have been negatively correlated with tax. The tax value is likely not a good indicator of the wealth in the area. I find it a bit hard to guess at what these correlations mean. One interesting value is that having higher proportion of blacks by town has a weak negative correlation with per capita crime rate.
Super bonus, 3D plots.
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)
# Plot with crime classes as color of the training data.
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)# Plot with clusters of k-means as color of the training data.
# First one needs to do k-means with 4 clusters to compare the methods.
km3D <-kmeans(boston_scaled, centers = 4)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km3D$cluster[ind])Comparison of the methods using eye-balling: the high crime rate cluster and third cluster of k-means are rather similar. The k-means clustering seems to work better as far as visuals go: less intermingling. The crime classes are more intermingled even to the point where low and medium high crime rate classes are interspersed. Cluster 1 and low somewhat match, but cluster 2 and 4 and medium high and medium low clusters do not match well. At least a one run of the file. Maybe not another. Note that you might need to left-click poke at the figures to see anything.